due to (REF ).
We can now apply known results on uniformly parabolic quasi-linear equations to (REF ).
In particular, from [1]}, Lemma 7.4 of Chapter II, page 119, we obtain the following.

Suppose statement (ii) of prop:iterative.scheme holds.
Then there exist constants \(\theta ,\eta \in (0,1)\)  depending only on \(N\) , \(c_1\) , \(c_2\) , \(m\)  and \(L\)  such that for any \(\bar{R}>0\)  with
\(Q^\theta (\bar{R}):=Q(\bar{R},\theta \bar{R}^2)\subseteq \bar{Q}_{n_*}\) 
